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The Non Equilibrium Steady State of Sparse Systems with Non Trivial Topology Daniel Hurowitz Department of Physics, Ben Gurion University, Beer Sheva Htotal = diag{En } − f (t){Vnm } + F · {Wnm } + HBath I Doron Cohen (BGU) [1,2] I Saar Rahav (Technion)[2] 1. D. Hurowitz and D. Cohen, Europhysics Letters 93, 60002 (2011) 2. D. Hurowitz, S. Rahav and D. Cohen, Europhysics Letters 98, 20002 (2012). “Sparsity” Htotal = diag{En } − f (t){Vnm } + F · {Wnm } + HBath Histogram of couplings n wn,n+1 Histogram 25 20 n+1 15 10 5 0 −4 −2 0 log(g) 2 ←− few decades −→ 4 wn+1,n = gn 2 gn = |Vn,n+1 |2 = couplings “sparsity” = log wide distribution of couplings Current vs. driving Driving ; Stochastic Motive Force ; Current Regimes: LRT regime, Sinai regime, Saturation regime |Current|, SMF 0 10 −2 10 −4 10 SMF |Current| ~est lrt/sat −6 10 −10 −5 10 I 10 ∼ 0 10 5 10 ε2 1 E∩ E w exp − 2 sinh N 2 2 Extent of the “Sinai regime” is determined by width of distribution of rates Master equation description of dynamics Htotal = diag{En } − f (t){Vnm } + F · {Wnm } + HBath Quantum master equation for the reduced probability matrix: dρ 2 = −i[H0 , ρ] − [V, [V, ρ]] + W β ρ ≡ Wρ dt 2 Stochastic rate equation: X dpn = wnm pm − wmn pn dt m The transition rates: wnm = β wnm + wnm wnm = wmn = gnm 2 = En −Em exp − TB Steady state equation: ρ̇ = Wρ = 0 β wnm β wmn The Stochastic Motive Force (SMF) If we had only a bath wnm En −Em = exp − wmn TB ε2=0.002 2.5 ε2=0.195 SMF potential 2 We define a “field” wnm E(x) ≡ ln wmn 1.5 1 0.5 0 −0.5 x −1 0 200 400 600 800 and “potentials” E(x1 ; x2 ) = Z x2 E(x)dx [potential variation] x1 n o E∩ ≡ maximum |E(x1 ; x2 )| [activation barrier] I E ≡ E(x)dx if no driving = 0 [SMF] With driving, E 6= 0. This means Q n wn,n+1 6= Q n wn+1,n . 1000 Current vs. driving Driving ; Stochastic Motive Force ; Current Regimes: LRT regime, Sinai regime, Saturation regime |Current|, SMF 0 10 −2 10 −4 10 SMF |Current| ~est lrt/sat −6 10 −10 −5 10 I 10 ∼ 0 10 5 10 ε2 1 E∩ E w exp − 2 sinh N 2 2 Extent of the “Sinai regime” is determined by width of distribution of rates Emergence of the “Sinai regime” Sinai [1982]: Transport in a chain with random transition rates. Assume transition rates are uncorrelated.√ ; build up of a potential barrier E∩ ∝ N . ; exponentially small current. But... we have telescopic correlations: En,n+1 ∼ ∆n ≡ (En −En+1 ) Yet... we have sparsely distributed couplings: wn,n+1 = gn 2 E ≈ − X n 1 ∆n 1 ∼ 1 + gn 2 TB TB 2 , 2 1/ √ , [±] N ∆, Build up may occur if gn are from a log-wide distribution. I ∼ 1 E∩ E w exp − 2 sinh N 2 2 2 < 1/gmax 2 > 1/gmin otherwise Beyond fluctutation dissipation phenomenology: Topological term in EAR formula Q̇ = Xh i β β w← p − w p ∆n − − → n n n n−1 −5 10 DB DB − (0) + TB E I TB T i DB h (gn 2 ) − (gn 2 )2 + Var(g)4 TB ≈ ≈ EAR n −10 10 total EAR excess EAR Topological term Current −15 10 −10 10 The EAR is correlated with the current. 0 10 ε2 The quantum mechanical steady state = X In→m = wmn pn − wnm pm ≡ tr(În→m ρ) wnm pm − wmn pn m În→m = |niwmn hn| − |miwnm hm| β β β În→m = |niwmn hn| − |miwmn hm| Quantum dpn dt Classical Stochastic 0.070 0.069 0.068 0.067 0.066 0.065 0.064 0.070 0.069 0.068 0.067 0.066 0.065 0.064 -5 0 5 Ε2 Quantum dρ dt În→m Jˆnm = −i[H0 , ρ] − [V, [V, ρ]] + W β ρ h i2 2 ˆnm = i J , V̂ = i |miVmn hn| − |niVnm hm| Current 0.1 2 0.001 10-5 10-7 10-8 10-5 0.01 10 Ε2 104 107 Summary of main results 1. Due to the sparsity of the perturbation matrix, the NESS is of glassy nature [1]. 2. An extension of the Fluctuation-Dissipation phenomenology has been proposed [1]. 3. A log-wide distribution of couplings is required in order to have a Sinai regime. 4. The topological term in the EAR is correlated with the current but sub-linear in driving intensity. 5. Novel saturation effect in the quantum model. 6. The quantum current operator in the reduced description includes off diagonal elements of the probability matrix. [1] D. Hurowitz and D. Cohen, Europhysics Letters 93, 60002 (2011). References and Acknowledgements 1. D. Hurowitz and D. Cohen, Europhysics Letters 93, 60002 (2011). 2. D. Hurowitz, S. Rahav and D. Cohen, Europhysics Letters 98, 20002 (2012). I Sparsity: Austin, Wilkinson, Prosen, Robnik, Alhassid, Levine, Fyodorov, Chubykalo, Izrailev, Casati I Energy absorption by sparse systems: Cohen, Kottos, Schanz, Wilkinson, Mehlig I Network theory: Schnakenberg, Zia, Hill I Sinai physics: Sinai, Derrida, Pomeau, Burlatsky, Oshanin, Mogutov, Moreau, Bouchard Acknowledgement: Bernard Derrida